On Simple Connectedness And Fundamental Groups Of Finitely Dimensional Algebras

The notion of simple connectedness came from topology , and the corresponding ones also appear in some subjects such as Analysis.It was given new meaning when introduced in the representation theory .There are close relations between the simple connectedness of finite-dimensional algebras and the covering theory . Simply connected algebra of representation-finite have been thoroughly studied ,since Bongartz and Gabrie studied it. But the knowledge of simply connected algebra with representation-infinite type is very poor, only some special class of it was carefully invesitigated. The notion of fundamental group also came from topology. There are close relations with simple connectedness in representation theory of algebras for fundamental group, as was in algebraic topology .Lower degree Hochschild cohomology group have a very concrete interpretation of classic algebraic structure , especially, there are inner connection between 1 Hochschild cohomology group and simply connected algebra. The relationship of simple connectedness and 1 Hochschild cohomology group is very clear for representation-finite algebras. Only one kind of representation-infinite algebras was. discussed in the paper. It gives some fundamental notations which will be used in this paper and some relevant background in chapter 1. In chapter 2, we got some conclusions for simple connectedness of minimal representation-infinite algebra. i. e for an algebra of minimal representation-infinite type with preprojective component, it is simply connected if and only if the vanishing of 1 Hochschild cohomology group;the same conclusion is true for a general algebra with minimal representation-infinite type.In chapter 3, we computed the fundamental group for hereditary algebra and other special cases, and studied thefundamental group under one point extension.